The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

Letters of UDAYPUR: A, D, P, R, U, U, Y (U repeated twice).
Words starting with a letter before U: letters A, D, P, R each give $\frac{6!}{2!} = 360$ arrangements $\Rightarrow 4 \times 360 = 1440$.
Fix U at position 1. Remaining: A, D, P, R, U, Y (all distinct).
Words starting with UA: $5! = 120$.
Fix UD at positions 1-2. Remaining: A, P, R, U, Y. No letter before A, so fix UDA.
Remaining: P, R, U, Y. Letters before Y: P, R, U give $3 \times 3! = 18$.
Fix UDAY. Remaining: P, R, U. No letter before P, fix UDAYP.
Remaining: R, U. Letter before U: R gives $1! = 1$.
Fix UDAYPU, last letter R: the word itself.
Rank $= 1440 + 120 + 18 + 1 + 1 = 1580$.